One of the main motivations for Quantum Field Theory after Dirac Equation is that the Dirac equation predicts negative energy states which leads to the ground state being unbounded which ultimately leads to the sea of negative energy electrons. It is said that the field theoretic approach cures it of this problem.
But every QFT book I have seen often defines the ground state $|0\rangle$ as:
$$ a|0\rangle=0 $$
but I can't see the basis for this definition ?
Of course, once I define the ground state as this it will automatically never lead to negative energy states? Why isn't this arbitrary? Why could I not do the same with Dirac Equation and define its ground state as such and hence remove all such negative energy states?